Question

The number of common roots among the 12^th and 30^th roots of unity is ?

• A. \(12\)
• B. \(9\)
• C. \(8\)
• D. \(6\)

Hint:

For \(n\)th and \(m\)th roots of unity, the common solutions are exactly the \(\gcd(n,m)\)th roots of unity.

Answer 1

The Correct Option is D

Step 1: Roots of unity definition.

The 12^th roots of unity are the complex solutions to \(z^{12} = 1\).

The 30^th roots of unity are the complex solutions to \(z^{30} = 1\).

Step 2: Common solutions.

A complex number is a common root of both equations if and only if it satisfies

\[ z^{12} = 1 \quad\text{and}\quad z^{30} = 1. \]

Equivalently, \(z\) is a root of unity whose order divides both 12 and 30.

Step 3: Use greatest common divisor.

The common roots are exactly the \(\gcd(12,\,30) = 6\)-th roots of unity. Hence there are 6 common roots.